The History of Mathematical Proof in Ancient Traditions

(Elle) #1

Reading proofs in Chinese commentaries 445


the outline of a problem asks to fulfi l. Th is detail indicates that, in ancient
China, algorithms may have been conceived as composed by combining a
sequence of algorithms which carry out a sequence of tasks, the comple-
tion of which was identifi ed as leading to the solution of a given kind of
problem. Th is corresponds quite well to the kind of reasoning Liu Hui has
been developing so far in the commentary we are reading.
Th e second word to be stressed is ‘also’. It refers to the fact that the same
argument was given earlier in the commentary, aft er problem 5.9, when Liu
Hui was deriving the algorithm for the volume of the cylinder from that of
the volume of the parallelepiped. Th is ‘also’ thus indicates that the proofs
are not carried out in isolation from each other, but rather in parallel with
each other – a fact that we have already stressed above.
In fact, aft er problem 1.33, devoted to computing the area of a circle, Liu
Hui had derived the values of 3 to 4 as corresponding lü s for the area of the
circle and that of the circumscribed square, respectively, from the values
of 3 to 1 for expressing the relationship between the circumference of the
circle and its diameter. In the commentary on problem 5.9, these values
were declared to allow the transformation of the volume of a cylinder into
that of the circumscribed parallelepiped. Th e same statement is made here,
and the geometrical assertion is followed by its translation into algorithms
(transformation 5): the same multiplication by 3 and division by 4 ensure
the transformation from the truncated pyramid with square base into the
truncated pyramid with circular base. As Liu Hui puts it:


Hence, if one multiplies by the lü of the circle, 3, and divides by the lü of the square,
4, one obtains the volume of the truncated pyramid with circular base.


As a consequence, at this point of his commentary, Liu Hui has deter-
mined a correct algorithm yielding the volume of the desired truncated
pyramid, which ends the fi rst line of argumentation. Algorithm 4 correctly
yields the value of the desired magnitude.


Multiplications Division Division Multiplication by 3
Sum by 9 by 3
Multiplication by h

C (^) i > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > [( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9]/3 > [[( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9]/3].3
C (^) s
Division by 4



[[[( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9]/3].3]/4


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